Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 0cld | Structured version Visualization version GIF version |
Description: The empty set is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 4-Oct-2006.) |
Ref | Expression |
---|---|
0cld | ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dif0 4334 | . . 3 ⊢ (∪ 𝐽 ∖ ∅) = ∪ 𝐽 | |
2 | 1 | topopn 21516 | . 2 ⊢ (𝐽 ∈ Top → (∪ 𝐽 ∖ ∅) ∈ 𝐽) |
3 | 0ss 4352 | . . 3 ⊢ ∅ ⊆ ∪ 𝐽 | |
4 | eqid 2823 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
5 | 4 | iscld2 21638 | . . 3 ⊢ ((𝐽 ∈ Top ∧ ∅ ⊆ ∪ 𝐽) → (∅ ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ ∅) ∈ 𝐽)) |
6 | 3, 5 | mpan2 689 | . 2 ⊢ (𝐽 ∈ Top → (∅ ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ ∅) ∈ 𝐽)) |
7 | 2, 6 | mpbird 259 | 1 ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 ∖ cdif 3935 ⊆ wss 3938 ∅c0 4293 ∪ cuni 4840 ‘cfv 6357 Topctop 21503 Clsdccld 21626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-top 21504 df-cld 21629 |
This theorem is referenced by: cls0 21690 indiscld 21701 iscldtop 21705 iccordt 21824 isconn2 22024 tgptsmscld 22761 mblfinlem2 34932 mblfinlem3 34933 ismblfin 34935 |
Copyright terms: Public domain | W3C validator |